Effective masses, electronic and optical properties of (111)-layered B-site deficient hexagonal perovskite Ba₅M₄O₁₅ (M = Ta, Nb): a DFT study using HSE06

Abstract

Equilibrium lattice parameters, electronic structures and optical properties of (111)-layered B-site deficient hexagonal perovskite Ba₅M₄O₁₅ (M = Ta, Nb) were studied by first-principles computations on the basis of density functional theory using the norm-conserving-type pseudo-potential technique and screened nonlocal exchange-correlation functional HSE06 as defined by Heyd, Scuseria, and Ernzerhof. The calculated band dispersions showed that Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are indirect band gap materials (A → G) with band gaps of 3.81 and 3.56 eV, respectively. The effective masses of photogenerated electrons and holes for Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ were evaluated in two principal directions at the G (Gamma) point. The Ta–O and Nb–O bonds in the MO₆ octahedral environments have polar covalent nature due to the p–d hybridization between O-2p and Ta-5d or Nb-4d orbitals. Since the valence and conduction bands of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ mainly consist of O-2p and Ta-5d or Nb-4d states, changes in the structure of the MO₆ octahedral units can be effective for the band gap energy and consequently photocatalytic activity of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅. The optical analysis revealed that the main peak of the imaginary part of the complex dielectric function of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ corresponds to the interband electronic transition from O-2p to Ta-5d or Nb-4d. Also, anisotropies in the effective masses of photogenerated charge carriers and static dielectric tensors of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ in an arbitrary crystallographic direction are presented. High photocatalytic activity of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ for hydrogen generation from water splitting and photodegradation of organic pollutants and/or dye molecules under UV light is related to the light effective masses of photogenerated charge carriers. For the efficient solar-energy conversion, the electronic band structures, such as band-edge position and band gap, of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ can be tuned by doping.

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Introduction

Photocatalytic degradation of organic pollutants and splitting of water into hydrogen using a heterogeneous photocatalyst has received considerable attention for environmental and energy goals. One of the most challenging issues in this field is the development of highly efficient and highly stable photocatalysts for a large-scale conversion of solar energy and water to hydrogen.¹ Perovskite and related materials have been demonstrated to be active for clean hydrogen production and direct water splitting.² Among them, (111)-layered B-site deficient hexagonal perovskite A₅M₄O₁₅ (A = Sr, Ba; M = Ta, Nb) are highly active photocatalysts for water splitting under ultra-violet (UV) illumination^3–5 and possess good microwave dielectric properties, including high relative permittivity, high quality factor, and low temperature coefficient of resonator frequency.^6–11 This type of perovskite belongs to the perovskite oxides with a general formula of AMO₃. When the sum of the formal charges of A and M cations are more than 6, a cation vacancy on the M sublattice occurs to grant the requirement of electroneutrality. Based on this rule, A₅M₄O₁₅ can be reduced to AM_0.8O₃ with a vacancy of 0.2 M cation per 1A cation. Thus, they are regarded as cation-deficient perovskite.^7,12

The A₅M₄O₁₅ compounds have a hexagonal symmetry and are crystallized in the space group of P̄3m1, in which each unit cell is composed of five close-packed layers containing one A and three O atoms, and the M atoms are located in octahedral holes between these layers. According to the electroneutrality requirement, one of the octahedral holes is empty, resulting in the loss of face sharing on the MO₆ sublattice and strong anharmonicity of these compounds.^7,11,13–16 Pentabarium tetratantalum oxide (Ba₅Ta₄O₁₅) and pentabarium tetraniobium oxide (Ba₅Nb₄O₁₅) containing closed-shell transition metal ions Ta⁵⁺/5d⁰ and Nb⁵⁺/4d⁰ are two important members of this family. Particularly, the MO₆ octahedral units are provenance of some important properties of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅. Srivastava et al.¹³ showed that the blue emission of Ba₅Ta₄O₁₅ and yellow emission of Ba₅Nb₄O₁₅ are due to a large relaxation in the electronic delocalization of the excited d⁰ configuration of the MO₆ octahedral units. Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ have also excellent microwave dielectric properties for optoelectronic applications, such as high permittivity 28 and 39, quality factor multiplied by resonance frequency 5700 × 5.55 and 5000 × 4.73 GHz, and temperature coefficient of resonance frequency 12 and 78 ppm °C⁻¹.⁶ Jawahar et al.¹⁷ suggested that the higher permittivity of Ba₅Nb₄O₁₅ is associated with its higher lattice anharmonicity.

From the view point of photocatalytic activity, Mukherji et al.¹⁸ pointed out that the MO₆ octahedral units are active sites for the photocatalytic activity of these materials. The photocatalytic activity of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ has been studied by a number of groups and demonstrated them to be highly active photocatalysts for water splitting under UV light. Depending on the employed synthesis routes, the experimental band gaps of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ were reported to be in the range of 3.90–4.50 eV ^{4,5,15,18–20} and 3.84–3.92 eV,^14,21,22 respectively. Since Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ have wide band gaps, these hexagonal perovskites can also serve as excellent starting materials for the fabrication of the cubic perovskite oxynitrides (BaTaO₂N and BaNbO₂N) with narrow band gaps.¹⁸

In our earlier work, we have demonstrated a contrasting effect of the Ta/Nb ratio in (111)-layered B-site deficient hexagonal perovskite Ba₅Nb_4−xTa_xO₁₅ (0 ≤ x ≤ 4) crystals grown by a flux method on visible-light-induced photocatalytic water oxidation activity of their oxynitride derivatives, BaNb_1−xTa_xO₂N (0 ≤ x ≤ 1).²³ In this work, we have comparatively studied the equilibrium lattice parameters, electronic structures and optical properties of (111)-layered B-site deficient hexagonal perovskite Ba₅M₄O₁₅ (M = Ta, Nb) by the first-principles computations on the basis of density functional theory using the norm-conserving-type pseudo-potential technique and screened nonlocal exchange-correlation functional HSE06 as defined by Heyd, Scuseria, and Ernzerhof.

Computational methodology

The atomic positions of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ were taken as initial geometries from the experimental crystal structures from our earlier work.²³ All calculations in this paper were carried out using the Cambridge Serial Total Energy Package (CASTEP),²⁴ which uses a plane wave expansion technology in reciprocal space. An interaction between the ionic core and valence electrons is typically represented by a nonlocal norm-conserving-type pseudo-potential (NCPP) with the frozen-core approximation²⁵ in order to allow calculations to be performed with the lowest possible cut-off energy with minimum contribution from the core region. The states of Ba (5s² 5p⁶ 6s²), Ta (5s² 5p⁶ 5d³ 6s²), Nb (4s² 4p⁶ 4d⁴ 5s¹), and O (2s² 2p⁴) electrons were treated as the valence electron configurations in the calculations. The scalar relativistic effects (SR) for heavy atoms were included in the construction of NCPP, while the nonscalar relativistic effects, spin–orbital coupling (SOC), is not considered. A kinetic energy cut-off value 850 eV was used for the plane wave expansions in reciprocal space. The self-consistent total energy calculations were performed using the screened nonlocal exchange-correlation functional HSE06 (Heyd–Scuseria–Ernzerhof).^26,27 The HSE06 exact exchange admixture comes from the perturbation theory arguments, and the screening parameter was chosen empirically.²⁸ A fast Fourier transformation (FFT) was set as 45 × 45 × 90 mesh, and the Monkhorst–Pack²⁹ scheme k-points grid sampling was set as 7 × 7 × 4 for the irreducible Brillouin zone in which the spacing of grid points was smaller than 0.03 Å⁻¹ for all calculations. The minimization algorithm of Broyden–Fletcher–Goldfarb–Shanno (BFGS)³⁰ was employed for geometry optimizations with total energy convergence tolerance 5 × 10⁻⁶ eV per atom. The structure and atoms was relaxed until the mean Hellmann–Feynman force acting on the atoms was less than 0.01 eV Å⁻¹. Within current setting other convergence parameters were as follows: self-consistent field tolerance 5 × 10⁻⁷ eV per atom, maximum stress 0.02 GPa, and maximum ionic displacement 5 × 10⁻⁴ Å. The separation between k-points in the band structure calculations was adopted by 0.001 Å⁻¹.

Results and discussion

The crystal structures of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ belong to the hexagonal A_mM_m−1O_3m perovskites with space group P-3m1 (no. 164). The conventional cells of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ contain three types of Ba atoms (1 atom with site of 1a and 2 + 2 atoms with site of 2d), two types of M atoms (2 atoms with site of 2c and 2 atoms with site of 2d), and three types of O atoms (3 atoms with site of 3e, and 6 + 6 atoms with site of 6i). The primitive cell of the hexagonal Ba₅M₄O₁₅ (M = Ta, Nb) is illustrated in Fig. 1a. The optimized crystallographic parameters, atomic positions, and structural parameters of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ under symmetry constrains are listed in Tables 1 and 2 and compared with the experimental values. The main difference between the theoretical and experimental values is due to certain intrinsic approximations in the exchange-correlation functional.^31,32 On the other hand it should be noted that experimental measurements have been done at 298 K while geometry optimizations are performed at zero temperature. Also, there are other important factors which may be present in the samples but have not been considered in the calculations, such as defects, impurities, etc. The bond angles of M₁–O₂–M₂ (Fig. 1b) in the corner-shared MO₆ octahedral units are deviated from 180°. The magnitude of the deviation of the M₁–O₂–M₂ bond angles from 180° directly affect the geometrical coupling of d⁰ metal ion polyhedra and is critically related to the excited state delocalization in the structure.¹³ Since the M₁–O₂–M₂ bond angles in Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are approximately equal, the difference between band gap energies of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ cannot be related to this parameter.

Fig. 1 (a) (111)-layered B-site deficient perovskite structure and primitive cell of the hexagonal Ba₅M₄O₁₅ (M = Ta, Nb) and (b) atom numbering in the MO₆ octahedral environment. The barium, tantalum/niobium, and oxygen atoms are represented by green, azure, and red spheres, respectively. The MO₆ octahedral units are shown in blue color.

Table 1 The optimized crystallographic parameters and final atomic positions of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ compared to the experimental values from ref. 23^a

Crystallographic parameters	Atom	Wyckoff position	x	y	z
a Experimental values are given in parenthesis.
Ba5Ta4O15 a = b = 5.9815 (5.7892) Å c = 12.2234 (11.8202) Å α = β = 90°; γ = 120°	Ba (1)	1a	0.0000 (0.0000)	0.0000 (0.0000)	0.0000 (0.0000)
	Ba (2)	2d	0.3333 (0.3333)	0.6667 (0.6667)	0.7851 (0.7872)
	Ba (3)	2d	0.3333 (0.3333)	0.6667 (0.6667)	0.4338 (0.4304)
	Ta (1)	2c	0.0000 (0.0000)	0.0000 (0.0000)	0.6836 (0.6859)
	Ta (2)	2d	0.3333 (0.3333)	0.6667 (0.6667)	0.1045 (0.1029)
	O (1)	3e	0.5000 (0.5000)	0.0000 (0.0000)	0.0000 (0.0000)
	O (2)	6i	0.1652 (0.1679)	−0.1652 (−0.1679)	0.1921 (0.1922)
	O (3)	6i	0.1688 (0.1641)	−0.1688 (−0.1641)	0.6109 (0.6159)
Ba5Nb4O15 a = b = 5.8526 (5.7946) Å c = 11.9395 (11.7876) Å α = β = 90°; γ = 120°	Ba (1)	1a	0.0000 (0.0000)	0.0000 (0.0000)	0.0000 (0.0000)
	Ba (2)	2d	0.3333 (0.3333)	0.6667 (0.6667)	0.7927 (0.7914)
	Ba (3)	2d	0.3333 (0.3333)	0.6667 (0.6667)	0.4285 (0.4272)
	Nb (1)	2c	0.0000 (0.0000)	0.0000 (0.0000)	0.6836 (0.6813)
	Nb (2)	2d	0.3333 (0.3333)	0.6667 (0.6667)	0.1050 (0.1041)
	O (1)	3e	0.5000 (0.5000)	0.0000 (0.0000)	0.0000 (0.0000)
	O (2)	6i	0.1702 (0.1707)	−0.1702 (−0.1707)	0.1913 (0.1935)
	O (3)	6i	0.1628 (0.1630)	−0.1628 (−0.1630)	0.6139 (0.6142)

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Table 2 The calculated Mulliken bond population, population ionicity, bond lengths (compared to the experimental values from ref. 23), octahedral distortion percentage (DP), and M₂–O₂–M₁ angle of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅

Sample	Bond (type)	Population	Ionicity	Bond length^a (Å)	DP (%)		M2–O2–M1 angle^a
					Cal.	Exp.
a Experimental values are given in parenthesis.
Ba5Ta4O15	Ta1–O2	0.22	0.971	2.288 (2.216)	15.39	18.48	170.01 (171.91)
	Ta1–O3	0.45	0.705	1.961 (1.841)
	Ta2–O1	0.39	0.790	2.147 (2.067)	4.91	5.00
	Ta2–O2	0.52	0.602	2.044 (1.966)
Ba5Nb4O15	Nb1–O2	0.13	0.998	2.282 (2.261)	21.01	21.77	171.02 (172.10)
	Nb1–O3	0.27	0.933	1.848 (1.817)
	Nb2–O1	0.24	0.957	2.104 (2.074)	7.70	6.52
	Nb2–O2	0.34	0.856	1.948 (1.943)

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In MO₆ octahedral units, there are two different M–O bond lengths. The distortion of crystallographically inequivalent MO₆ octahedral units can be calculated by:³³

The M–O bond lengths and percentage values of the octahedral distortions showed that the MO₆ units in Ba₅Nb₄O₁₅ are more distorted in comparison to that in Ba₅Ta₄O₁₅. Thus, higher permittivity of Ba₅Nb₄O₁₅ can be due to the relatively higher lattice anharmonicity of Ba₅Nb₄O₁₅.⁷ The valence and conduction bands of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are mainly composed of the orbitals of the MO₆ units; therefore, the generated dipole moment due to the octahedral distortions can play an important role in the photocatalytic activity of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅.

The Mulliken bond population as a measure of spatial charge density between bonding atoms was calculated to understand the bonding nature of each M–O bonds in the MO₆ octahedral units. The degree of covalence nature of a bond can be described using population ionicity (P_i) from the following equation:³⁴

In the mentioned equation P_c = 1 indicates the bond population for a purely covalent as reference, P is the bond population of studied bond, and 0 ≤ P_i ≤ 1. Upper and lower limits of P_i correspond to the purely covalent and purely ionic bonds, respectively.

The calculated Mulliken bond population and population ionicity of each M–O bond indicate that all M–O bonds in the MO₆ octahedral units are polarized covalent (Table 2). A comparison between the structurally similar M–O bonds in Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ shows that the Nb–O bonds have higher population iconicity and consequently the Nb–O bonds are more polar than the Ta–O bonds. The calculated Mulliken and Hirshfeld charges of the O and Ta/Nb atoms (Table 3) revealed that in the Nb–O bonds the shared electrons are highly polarized from metal ion to O atom with respect to the Ta–O bonds, which is in agreement with higher values of population ionicity of the Nb–O bonds. The cross-sections of the bonding charge densities of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ crystals in the base plane of the MO₆ octahedral units are sketched (Fig. 2) to better understand the nature of the M–O bonds, and it is clear that the charge density contours between the M and O atoms are highly polarized toward O atom, and the contours are mainly concentrated to the O atom, suggesting a polarized covalent bonding nature for the M–O bonds.

Table 3 Mulliken and Hirshfeld atomic charges of O and Ta/Nb of MO₆ octahedra units

Sample	Atom	Hirshfeld charge	Mulliken charge
Ba5Ta4O15	Ta1	0.68	1.62
	Ta2	0.69	1.64
	O1	−0.31	−0.95
	O2	−0.30	−0.94
	O3	−0.31	−0.90
Ba5Nb4O15	Nb1	0.80	1.73
	Nb2	0.81	1.89
	O1	−0.35	−1.02
	O2	−0.33	−0.98
	O3	−0.33	−0.95

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Fig. 2 Total charge density distribution of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ in the base plane of the MO₆ octahedral units.

The calculated band structures of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ along high symmetry directions in the first Brillouin zone (Fig. 3a) are presented in Fig. 3b. The top of the valence band (VB) was chosen as zero of energy, and the Fermi energy level was set at the valence band maximum (VBM). The Bradley–Cracknell notation was used for the high-symmetry points. The coordinates of the special points of the Brillouin zone in terms of unit vectors of the reciprocal lattice are G (0, 0, 0), M (0, 0.5, 0), K (−0.333, 0.667, 0), A (0, 0, 0.5), L (0, 0.5, 0.5), and H (−0.333, 0.667, 0.5). The band dispersion structures clearly indicate that in Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ the valence band maximum (VBM) and conduction band minimum (CBM) are located at different k-points, implying that these are indirect band gap materials (A → G) with the electronic band gaps of 3.81 and 3.56 eV, respectively. The calculated band gap values of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are in good agreement with experimental values from ultraviolet-visible (UV-Vis) diffuse reflectance spectra which are 3.98 and 3.59 eV for Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅, respectively.²³ Since Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are isostructural materials, a smaller band gap of Ba₅Nb₄O₁₅ compared to that of Ba₅Ta₄O₁₅ is due to the higher effective electronegativity of Nb⁵⁺ ion.³⁵

Fig. 3 (a) First Brillouin zone of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ in the reciprocal lattice. The red line corresponds to the path of the band structure diagram, and g1, g2, and g3 denote the reciprocal vectors. The calculated band structures of Ba₅Ta₄O₁₅ (b) and Ba₅Nb₄O₁₅ (c). The Fermi energy (E_F) is set to be 0 eV and is marked by a horizontal dashed line.

The photocatalytic activity of a photocatalyst is related to the effective masses of photogenerated charge carriers which are related to the band dispersion at the extreme points of the forbidden band edge as:^36,37

where E(k⃑) is the energy at wave-vector k in the band, E(k⃑₀) is a constant giving the edge of energy of the band, , , and are effective masses of electrons or holes in the [100], [010], and [001] directions, respectively, and m_e is the mass of free electron. Owing to the hexagonal symmetry of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅, the effective masses of photogenerated charge carriers along the [100], [010], and [001] directions are m^*_x = m^*_y ≠ m^*_z. By parabola fitting of the band dispersions of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅, the effective masses of electrons at the bottom of CB, and holes on the top of VB were calculated. Meanwhile, the spherical averaged and three-dimensional contour plots of the effective masses were also studied at the gamma point using the following expressions:³⁸where θ and φ are polar and azimuthal angles in spherical coordinates, respectively. S_ij tensor is related to the inverse of effective mass by:³⁶where , and . The calculated values of the effective masses of electrons and holes in three principal directions and spherical averaged of electrons and holes for Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are given in Table 4.

Table 4 The calculated effective masses of photogenerated electrons and holes in principal directions and spherical averages of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅

Sample	Electron			Hole
	[010]	[001]	Average	[010]	[001]	Average
Ba5Ta4O15	0.012	0.017	0.013	0.030	0.015	0.024
Ba5Nb4O15	0.012	0.007	0.010	0.152	0.120	0.140

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The effective masses of photogenerated electrons at the bottom of the CB and photogenerated holes on the top of VB for Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are smaller than those of the conventional semiconductor photocatalysts. For example, in rutile TiO₂, the effective masses of electrons along the [100] and [001] directions are ∼2–4 and ∼10–15, respectively.³⁹ Previously, some oxide and non-oxide compounds have also been reported to have very low effective masses of electrons. The effective masses of electrons for cubic BaSnO₃ and Sr_0.2Ba_0.8SnO₃ calculated using GGA-DFT method were found to be in the range of 0.03–0.09.⁴⁰ The effective masses of electrons for Cd₂SnO₄ and indium-doped Cd₂SnO₄ were observed experimentally to be between 0.05–0.09.^41,42 Also, the effective masses of electrons for InSb, GaS, GaInAs, Ga_0.7In_0.3N_0.004As_0.996, and Ga_0.7In_0.3N_0.01As_0.99 were estimated to be 0.013, 0.067, 0.054, 0.07, and 0.08, respectively.^43–45 The very small effective masses of holes or electrons suggest that the band at the VBM or CBM has very high curvature and very low density of states at its maximum or minimum, respectively.

The very low effective masses of electrons and holes for Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ imply that the photogenerated charge carriers have more possibility to participate in the photocatalytic process before recombination. The higher photocatalytic activity of Ba₅Ta₄O₁₅ in comparison to Ba₅Nb₄O₁₅ ^19,20,22 can be justified from its photogenerated charge carriers with lighter effective masses. Spherical averaged effective mass values of electrons and holes indicate that Ta–O bonds in Ba₅Ta₄O₁₅ show stronger covalent character than Nb–O bonds in Ba₅Nb₄O₁₅. The three-dimensional contours of the effective masses of photogenerated carries and their projections at (010) crystal plane are shown in Fig. 4 and 5, respectively. As implied by tensor elements, the effective masses of electrons and holes in Ba₅Ta₄O₁₅ and the effective masses of electrons in Ba₅Nb₄O₁₅ show strong anisotropic contours.

Fig. 4 Three-dimensional contour plots of the effective masses of photogenerated electrons and holes of (a) Ba₅Ta₄O₁₅ and (b) Ba₅Nb₄O₁₅.

Fig. 5 Projection of anisotropy in the effective masses of electrons (solid line) and holes (dashed line) at (010) crystal plane in the first irreducible Brillouin zone for Ba₅Ta₄O₁₅ (blue line) and Ba₅Nb₄O₁₅ (red line). The high symmetry k-points are shown by the Bradley–Cracknell notation.

In order to analyze the composition of the calculated electronic bands, the total, site- and angular-projected density of states (DOS) diagrams of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are illustrated in Fig. 6. The band structures of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ from −10 to −12 eV are essentially contributed by Ba-5p states with a very small admixture of O-2p and Ta-5d or Nb-4d. The valence bands of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ lie between −7 to 0 eV, which are mainly from O-2p orbital overlapping with Ta-5d or Nb-4d orbitals. The p–d hybridization between O and Ta or Nb atoms leads to the formation of highly polarized covalent M–O bonds.

Fig. 6 Total, site- and angular-projected electronic density of states (DOS) of (a) Ba₅Ta₄O₁₅ and (b) Ba₅Nb₄O₁₅. Color legends: total DOS in black, s orbitals in blue, p orbitals in red, and d orbitals in green. The Fermi energy (E_F) is set to be 0 eV.

The projected electronic wave function (PEWF) contour plots at the top of the VB between −3 to 0 eV in the base plane of the MO₆ octahedral units of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are plotted in Fig. 7. The PEWFs of the high energy set of valence band show the p orbital lobes on O atoms, indicating that this region is mainly composed of O-2p orbital. As can be seen, the charge density around the Ta/Nb atoms is negligible. This is a natural result because Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ contain closed-shell transition metal ions with octahedrally coordinated Ta⁵⁺/5d⁰ and Nb⁵⁺/4d⁰ configurations. The conduction bands of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are essentially contributed by Ta-5d or Nb-4d orbitals with a minor admixture of other orbitals. Therefore, the electron transfer in Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ upon illumination occurs from O-2p orbital to the Ta-5d or Nb-4d orbitals.

Fig. 7 Projected wave function contour plots of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ in the base plane of the MO₆ octahedral units at the top of the VB between −3 to 0 eV.

Since the valence and conduction bands of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are mainly formed by O-2p and Ta-5d or Nb-4d orbitals, the presence of a local field due to the octahedral distortions at the MO₆ units can induce changes in the band structure, affect the band gap, and play a significant role in the photocatalytic activity of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅.

The real ε₁(ω) and imaginary ε₂(ω) parts of the complex dielectric function of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ were calculated with polycrystalline geometry and shown in Fig. 8a. The features of the real and imaginary parts of dielectric function along the polycrystalline direction are similar. The real part of dielectric function at zero energy indicates that the static dielectric constants for Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are 2.53 and 2.66, respectively. The real part of dielectric function in the energy range from 7 to 12 eV is negative because of damping of electromagnetic wave.⁴⁶ Also, zero values indicate that the longitudinally polarized waves are possible. Since Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are non-metal, the imaginary part of the complex dielectric function is related to the interband optical transitions between the different special k points at the first irreducible Brillouin zone. The main peaks of the imaginary parts of the complex dielectric function of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are located at 7.21 and 6.83 eV, respectively. This peak corresponds to the interband electronic transition from O-2p → Ta-5d or Nb-4d within the octahedral tantalate/niobate groups. In fact optical absorption of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ is due to charge transfer transitions within the MO₆ units. Meantime, the three-dimensional contour plots of static optical dielectric tensors along three polarization directions were studied. As the crystal structures of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ belong to the space group P̄3m1, their static optical dielectric tensors along three polarization directions can be written as:⁴⁷

Fig. 8 (a) The real (solid line) and imaginary (dashed line) parts of the dielectric function of Ba₅Ta₄O₁₅ (blue line) and Ba₅Nb₄O₁₅ (red line). The real (solid line) and imaginary (dashed line) parts of the dielectric function of (b) Ba₅Ta₄O₁₅ and (c) Ba₅Nb₄O₁₅ along the [010] (blue line) and [001] (red line) polarization vectors.

The real and imaginary parts of the dielectric function of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ along the [010] and [001] polarization vectors were calculated and shown in Fig. 8b and c, respectively. The bands of the real and imaginary parts of dielectric functions of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are contributed by the both real and imaginary parts of polarization directions [010] and [001]. The calculated nonzero diagonal elements of optical dielectric tensor for Ba₅Ta₄O₁₅ at the frequency approaching zero along the [010] and [001] directions are 2.63, and 2.33 and for Ba₅Nb₄O₁₅ are 2.79 and 2.39, respectively. As illustrated in Fig. 8b and c, the intensities of the main peaks of the real part of the dielectric function of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ along the [010] and [001] polarization vectors are different. Also, the features of the real and imaginary parts of the dielectric constant along the [010] and [001] polarization vectors and along the polycrystalline direction are similar.

The anisotropy in static optical dielectric constant in an arbitrary crystallographic direction was calculated using the following equation:

where λ_ij represents the matrix of directional cosines. The above-given equation can be simplified as:⁴⁷

ε(θ, φ) = ε_[100] sin²(θ)cos²(φ) + ε_[010] sin²(θ)sin²(φ) + ε_[001] cos²(θ) (9)

As shown in Fig. 9, the three-dimensional contour plots of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ do not evidence a strong anisotropy in static dielectric tensors.

Fig. 9 Three-dimensional contour plots of static optical dielectric tensors of (a) Ba₅Ta₄O₁₅ and (b) Ba₅Nb₄O₁₅.

The positions of valence and conduction bands of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ and the redox potentials of an adsorbate are effective parameters on the capabilities of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ to undergo photoinduced electron transfer to the adsorbed species on theirs surfaces. The relevant potential level of the acceptor and donor species is thermodynamically required to be more positive than the conduction band position (E_CB) and more negative than the valence band position (E_VB). Thus, the relative dispositions of the VB and CB potentials of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ according to their absolute electronegativities were calculated with respect to the normal hydrogen electrode (NHE) using the following equations:^48–53

E_VB = χ − E_e + 0.5 E^Optical_g (10)

E_CB = E_VB − E^Optical_g (11)

where E_e is the energy of free electrons on the hydrogen scale (approximately 4.50 eV) and χ is the absolute electronegativities of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅, which are the geometric mean of the electronegativities of the isolated component atoms:where P is the number of atoms in the studied materials and k = 1, 2, 3, …, P. According to the experimental electronegativity values (Ba = 2.40 eV, Ta = 4.11 eV, Nb = 4.00 eV, and O = 7.54 eV),⁵⁴ the absolute electronegativities of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are 5.37 and 5.34 eV, respectively. It is necessary to mention that in the calculated absolute electronegativities the effects of crystal structure and surface polarization are omitted. The positions of the VB and CB potentials of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ with respect to vacuum scale and NHE are illustrated in Fig. 10. As the valence band positions of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ are deep enough with respect to the redox potential of water oxidation (H₂O/O₂: 1.23 V), doping with an anionic element such as nitrogen having a higher p-orbital energy than oxygen can shift the valence band upward.¹⁸ Likewise, their conduction band positions are high enough with respect to the redox potential of hydrogen generation (H⁺/H₂: 0 V); therefore, doping with a cationic element such as W and Mo, which can add one extra electron to the system, can shift downward the conduction bands of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅, respectively. However, monodoping can create impurity states in the band gap that can act as a recombination center.⁵⁵ Therefore, the band gaps of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ can be reduced to suitable values (<3 eV) for the efficient solar-energy conversion without creating recombination centers by co-doping (W–N and Mo–N), respectively.

Fig. 10 Relative dispositions of the VB and CB potentials of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ with respect to vacuum scale and NHE.

Free from high band gaps of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ as a weak point, the studied compounds are suitable for photocatalytic hydrogen generation from water splitting under UV light. Also, the VB and CB positions of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ indicate that photogenerated electrons and holes can produce active species to decompose various organic pollutants and dye molecules under UV light.

Conclusions

To conclude, the equilibrium lattice parameters, electronic structures and optical properties of (111)-layered B-site deficient hexagonal perovskite Ba₅M₄O₁₅ (M = Ta, Nb) were investigated by the first-principles computations on the basis of density functional theory and screened nonlocal exchange-correlation functional HSE06. It was found that the NbO₆ octahedral units are more distorted than the TaO₆ units. The calculated Mulliken bond populations and cross-sections of the bonding charge densities showed that the Ta–O and Nb–O bonds in the corner-shared octahedral structures of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ have polar covalent nature. The band structure analysis revealed that Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ have indirect band gaps (A → G) of 3.81 and 3.56 eV, respectively. The smaller band gap of Ba₅Nb₄O₁₅ compared to that of Ba₅Ta₄O₁₅ is due to the higher effective electronegativity of Nb⁵⁺ than Ta⁵⁺. The effective masses of photogenerated electrons and holes in Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ were found to be very small with the exception for the effective masses of holes in Ba₅Nb₄O₁₅. The analysis of the density of states indicates that the polar covalent nature of the Ta–O or Nb–O bonds arises from the p–d hybridization between O-2p and Ta-5d or Nb-4d orbitals. Also, O-2p and Ta-5d or Nb-4d orbitals contribute to the formation of the valence and conduction bands of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅. Changes in the structures of the TaO₆ or NbO₆ octahedral units can affect the band gap energy and photocatalytic activity of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅. The main peak of the imaginary part of the complex dielectric function of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ corresponds to the interband electronic transition from O-2p to Ta-5d or Nb-4d. The static dielectric tensors of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ along the polarization vectors did not show a strong anisotropy. The positions of the valence and conduction bands revealed that the electronic structures of Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅ can be tuned by co-doping for the efficient solar-energy conversion. The studied compounds (Ba₅Ta₄O₁₅ and Ba₅Nb₄O₁₅) are suitable for photocatalytic hydrogen generation from water splitting and decomposition of various organic pollutants and dye molecules under UV light.

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